问题
若要求函数 $z$ $=$ $f(x, y)$ 在 $\varphi(x, y)$ $=$ $0$ 条件下的极值，且已经构造出了如下的拉格朗日函数：$F(x, y)$ $=$ $f(x, y)$ $+$ $\lambda$ $\varphi(x, y)$
则，根据拉格朗日乘数法，还需要构造以下哪个选项中的方程组并计算才可能得出与极值对应的驻点 $(x_{0}, y_{0})$ ?
选项
[A]. $\left\{\begin{array}{l}f_{x}^{\prime}(x, y)-\lambda \varphi_{x}^{\prime}(x, y)=0, \\ f_{y}^{\prime}(x, y)-\lambda \varphi_{y}^{\prime}(x, y)=0, \\ \varphi(x, y)=1. \end{array}\right.$[B]. $\left\{\begin{array}{l}f(x, y)+\lambda \varphi_{x}^{\prime}(x, y)=0, \\ f(x, y)+\lambda \varphi_{y}^{\prime}(x, y)=0, \\ \varphi(x, y)=0. \end{array}\right.$
[C]. $\left\{\begin{array}{l}f_{x}^{\prime}(x, y)+\lambda \varphi_{x}^{\prime}(x, y)=0, \\ f_{y}^{\prime}(x, y)+\lambda \varphi_{y}^{\prime}(x, y)=0, \\ \varphi(x, y)=0. \end{array}\right.$
[D]. $\left\{\begin{array}{l}f_{x}^{\prime}(x, y)+\lambda \varphi_{x y}^{\prime \prime}(x, y)=0, \\ f_{y}^{\prime}(x, y)+\lambda \varphi_{y x}^{\prime \prime}(x, y)=1, \\ \varphi(x, y)=0.\end{array}\right.$
$\left\{\begin{array}{l}f_{x}^{\prime}(x, y)+\lambda \varphi_{x}^{\prime}(x, y)=0, \\ f_{y}^{\prime}(x, y)+\lambda \varphi_{y}^{\prime}(x, y)=0, \\ \varphi(x, y)=0. \end{array}\right.$